Ž³ o ò q ©T ô p §# aÆ k Ó À W ¥ # a ¹ Å  Œƒ »ì ÅX ì Ä Ä Z ØÊ Ý w Š ¹ Å ­ Žä à Å4 “ Ö «“ Ó Þ

­

Ž³  o ò q ©T  ô p §# aÆ k Ó À W ¥ # a ¹ Å  Œƒ »ì ÅX ì Ä Ä Z ØÊ Ý w Š ¹ Å ­ Žä à Å4  “ Ö «“ Ó Þ



¡) o £ Ó

Ø 

æ· ¡ ¤ @ /† < Ɠ § Ó ü t o † < Æõ  x 9  BK21 Ó ü t o  á Ԗ ÐÕ ªÏ þ ›, ' õ AÅ Ò 361-763

(2012¸   3 Z 4 14{ 9  ~ à Î6 £ §, 2012¸   4 Z 4 5{ 9  à º& ñ ‘ : r ~ à Î6 £ §, 2012¸   5 Z 4 7{ 9  > F  S X ‰& ñ )

Prudnikov _  & h ì  r õ  / å L à º\  › ' a ô  Ç  ½ ™× ¼· ¡ ¤ õ  Gradshteynõ  Ryzhik_  & h ì  r, / å L à º x 9  Y  L \  › ' a ô  Ç Ã º³ ð

\


š ¸  H R

π 0

R

2π

0

f (a sin θ cos φ + b sin θ sin φ + c cos θ) sin θdθdφ _  ³ ð€   & h ì  r› ' a > d ” `  ¦ Ä »• ¸ % i  .

s

 3 l q& h `  ¦ 0 AK " f B Ä º 4 Ÿ ¤ ¸ ú šô  Ç Rodrigues_   r„  d ”  @ /’   ý a³ ð» ¡ ¤`  ¦ ¿ º     r„  `  ¦ r & " f & h ì  r s  ç ß –é ß – K

 t • ¸2 Ÿ ¤ ë ß –[ þ t # Q" f Ä »• ¸ % i  . Õ ªo “ ¦ s  & h ì  r s  : Ÿ x > % i † < Æ_  # Œ Q ë  H ] j\  # Qb  G>  6 £ x6   x ÷ &  H t \  ¦ ˜ Ð

% i  .

Ù þ

˜d ” # Q: { 9 ^ ‰y Œ • & h ì  r,  r„   ' Ÿ § > =, “ ¦„   Û ¼— 2 ;> 

An Angular Integral Involving a Scalar Product and Its Application to Classical Spin Systems

Suhk Kun OH ^∗

BK21 Physics Program and Department of Physics, Chungbuk National University, Cheongju, 361-763

(Received 14 March 2012 : revised 5 April 2012 : accepted 7 May 2012)

A surface integral formula R

π 0

R

2π

0

f (a sin θ cos φ + b sin θ sin φ + c cos θ) sin θdθdφ compiled without derivation in Integrals and Series by Prudnikov, Brychkov and Marichev and in Table of Integrals, Series and Products by Gradshteyn and Ryzhik is explicitly derived through two coordinate rotations instead of using the complicated Rodrigues rotation formula. We show how this integral can be used to investigate of many statistical physics problems.

PACS numbers: 05.90.+m, 05.10.-a, 02.10.Yn

Keywords: Angular integral, Rotation matrix, Classical spin systems

I. " e ] Ø

“

¦„   Û ¼— 2 ;> _  ¨ î + þ A: Ÿ x > % i † < Æ& h  $ í | 9 `  ¦ · ú ˜ ˜ Ðl  0 AK 

"

f  H ì  r C † < Êà º\  ¦ > í ß – # Œ    H X < s M : 2 " é ¶ ½ ¨³ ð€  



© œ_  & h ì  r`  ¦ à º' Ÿ ½ + É € 9 כ ¹ Ò q t|    [1–5]. s  Qô  Ç & h ì  r×  æ _

 
 Prudnikov et al. [6]õ  Gradshteyn et al. [7]\ 

∗

E-mail: [email protected]

z 

´ 2 ;  6 £ § _  & h ì  r› ' a > d ” s  .

Z π 0

Z 2π 0

f (a sin θ cos φ + b sin θ sin φ + c cos θ) sin θdθdφ

= 2π Z 1

−1

f ( p

a ² + b ² + c ² t)dt (1) s

 & h ì  r› ' a > d ” “ É r ¾ º Ä »• ¸Ù þ ¡  H t   H µ 1 ß) €4 R e ” t  · ú §“ ¦ Õ

ª   õ ë ß – é ß –í  H y  0 A_  Õ þ ˜[ þ t \  z  ´ 9e ”  .   " f s  & h  ì

 r› ' a > d ” `  ¦ é ß –í  H y  Ä »• ¸   H  כ “ É r Õ ª  ^ ‰ ë ß –Ü ¼– Ð < É ª p 

-454-

Fig. 1. (Color online) Vector ~ A and ˆ r in the xyz- coordinate system

e ”

  H { 9 s  “ ¦ ½ + É Ã º e ” t ë ß – s  כ `  ¦ é ß –> Z > – Ð ½ ¨^ ‰& h Ü ¼

–

Ð Ä »• ¸   H כ “ É r à º† < Æ x 9  Ó ü t o “ §¹ ¢ ¤ \ • ¸ _ p  e ”  “ ¦



« Ñ  ) a  .  8Ô  ¦ # Q s  & h ì  r`  ¦ # Q‹ "  Ó ü t o  ë  H ] j\  6 £ x6   x`  ¦ ½ + É Ã

º e ”   H t \  ¦ ¶ ú ˜( R˜ Ѝ  H  כ “ É r  8¹ ¡ ¤ _ p e ”   H { 9 s  “ ¦ ^  ¦ Ã

º e ”  . ‘ : r  7 Hë  H \ " f  H s  כ [ þ t`  ¦  [ jy   À Ò# Q ˜ Г ¦  ô

 Ç .

II. X ì Ä Ä Z Ø8 ý 4  ˜ m

d ”

 (1)`  ¦ Ä »• ¸ l  0 AK " f 7 ˜' 

A = aˆ ~ x + bˆ y + cˆ z (2) ü

< é ß –0 A 7 ˜'  ˆ

r = sin θ cos φˆ x + sin θ sin φˆ y + cos θˆ z (3)

\

 ¦ • ¸{ 9   . Õ ª Q€   d ”  (1)`  ¦  6 £ § õ  ° ú  s  ³ ð‰ & ³½ + É Ã º e ” 



.

Z π 0

Z 2π 0

f ( ~ A · ˆ r) sin θdθdφ = Z

f ( ~ A · ˆ r)dΩ. (4)

#

Œl " f dٍ  H { 9 ^ ‰y Œ • כ ¹™ è(solid angle element)s  .

Mathews ü < Walker [8]\  Å Ò# Q4 R e ”   H @ /g A$ í | 9  ~ ½ ÓZ O 

`

 ¦ s 6   x # Œ Fig. 1_  ˆr`  ¦ z-» ¡ ¤ \  { 9 u  • ¸2 Ÿ ¤  r„  r & 

˜

Ð . s X O >  l  0 AK " f  H €  $  ˆr`  ¦ z-» ¡ ¤ \  @ /K " f y Œ •

•

¸ −φë ß –
p u  r„  r †    6 £ § \  y-» ¡ ¤ \  @ /K " f y Œ •• ¸ −θ ë ß –

p

u  r„  r v €    ) a  . { 9 ì ø Í& h Ü ¼– Ð 7 ˜'  ˆ ω = (ω x , ω y , ω z ) \  _

K " f " î r   ) a “ ¦& ñ » ¡ ¤ \  @ /K " f y Œ •• ¸ θ ë ß –
p u  r„  r & Å Ò



 H  r„  ' Ÿ § > =`  ¦ ½ ¨   H ´ òÖ  ¦& h “   ~ ½ ÓZ O “ É r Rodrigues  r„   d ”

 (Rodrigues’ rotation formula) [9–11]`  ¦  6   x   H  כ s 



. s  d ” Ü ¼– Ð Â Ò'  % 3 # Qt   H  r„  ' Ÿ § > =“ É r  A ü < ° ú   .

R ω ˆ (θ) = e ^{ωθ ˆ}

= I + ˆ ω sin θ + ˆ ω ² (1 − cos θ)

=





cos θ + ω ² _x (1 − cos θ) ω _x ω _y (1 − cos θ) − ω _z sin θ ω _y sin θ + ω _x ω _z (1 − cos θ) ω z sin θ + ω x ω y (1 − cos θ) cos θ + ω ² _y (1 − cos θ) −ω x sin θ + ω y ω z (1 − cos θ)

−ω y sin θ + ω x ω z (1 − cos θ) ω x sin θ + ω y ω z (1 − cos θ) cos θ + ω _z ² (1 − cos θ)



 .

(5)

0

A\ " f ƒ  / å L ô  Ç z-» ¡ ¤ \  @ /ô  Ç ' Í   P :  r„  “ É r

R z ˆ (−φ) =





cos φ sin φ 0

− sin φ cos φ 0

0 0 1



 (6)

_

  r„  ' Ÿ § > =\  _ K " f l Õ ü t ) a  .Õ ªo “ ¦ D h– Ðî  r y-» ¡ ¤ \ 

@

/ô  Ç ¿ º  P :  r„  “ É r

R y ˆ

⁰

(−θ) =





cos θ 0 − sin θ 0 1 0 sin θ 0 cos θ



 (7)

_

  r„  ' Ÿ § > =\  _ K " f l Õ ü t ) a  . 0 A_  ¿ º ' Ÿ § > =`  ¦   ½ + Ë 

#

Œ 7 ˜'  ~ A \  & h 6   x €  

A ~ ⁰ = (a cos θ cos φ + b cos θ sin φ − c sin θ)ˆ x +(−a sin φ + b cos φ)ˆ y

+(a sin θ cos φ + b sin θ sin φ + c cos θ)ˆ z (8)

  ) a  . s  כ õ  ˆ r ⁰ = ˆ z\  ¦`  ¦ Û ¼º ú ˜ Y  L`  ¦ €   A ~ ⁰ · ˆ r ⁰ = a sin θ cos φ + b sin θ sin φ + c cos θ

= ~ A · ˆ r (9)

e ”

`  ¦ · ú ˜ à º e ”  . 7 £ ¤, Û ¼º ú ˜ Y  L _  ° ú כ“ É r  r„  \  @ /K " f Ô  ¦



 | ¾ Ós  . Õ ªo “ ¦ ~ A ⁰ _  ß ¼l \  ¦ ½ ¨K  ˜ Ѐ  

| ~ A ⁰ | = p

a ² + b ² + c ² (10) s

Ù ¼– Ð

A ~ ⁰ · ˆ r ⁰ = | ~ A ⁰ || ˆ r ⁰ | cos Θ

= | ~ A ⁰ | cos Θ

= a sin θ cos φ + b sin θ sin φ + c cos θ (11)

– Ð Â Ò' 

cos Θ = 1

√ a ² + b ² + c ² (a sin θ cos φ+b sin θ sin φ+c cos θ) (12)

\

 ¦ % 3   H  . Õ ªo  # Œ Z

f ( ~ A · ˆ r)dΩ = Z

f ( ~ A ⁰ · ˆ r ⁰ )dΩ ⁰

= Z π

0

Z 2π 0

f (| ~ A ⁰ | cos Θ) sin ΘdΘdΦ

= 2π Z π

0

f ( p

a ² + b ² + c ² cos Θ) sin ΘdΘ

= 2π Z 1

−1

f ( p

a ² + b ² + c ² t)dt (13)

e ”

`  ¦ 7 £ x" î >   ) a   [12]. s  ~ ½ Ó& ñ d ” _   t } Œ • ×  ¦ \ " f  H t ≡ sin Θ\  ¦ ‚  × þ ˜ # Œ & h ì  r`  ¦ u  ¨ 8 Š % i  .

III. X ì Ä Ä Z Ø8 ý “ Ö «“ Ó Þ

s

] j 0 A_  & h ì  r s  s 6   x ÷ &  H ˜ Ðl [ þ t`  ¦ ¶ ú ˜( R˜ Ðl – Ð ô  Ç



.

1. V ê s„ ÆX c l “ Ó ÞT  ° n Þ À W ¥ w Š ¹ Å ­ Žä à Å4 

&

ñ  l  © œ_  % ò † ¾ Ó A " f N> h_  " f– Ð  © œ  ñ Œ •6   x`  ¦ t 

· ú

§  H “ ¦„   Û ¼— 2 ;[ þ t“ É r  6 £ § Hamiltonian \  _ K " f l Õ ü t ) a



.

H = −~h ·

N

X

i=1

−

→ s i ,

= −h _x

N

X

i=1

s ^x _i − h _y

N

X

i=1

s ^y _i −

N

X

i=1

s ^z _i . (14)

#

Œl " f h ^x , h y , h z   H y Œ •y Œ • x-~ ½ ӆ ¾ Ó, y-~ ½ ӆ ¾ Ó, z-~ ½ ӆ ¾ Ó & ñ  l 



© œ_  ß ¼l s  . ¢ ¸ô  Ç s ^x _i = sin θ i cos φ i , s ^y _i = sin θ i sin φ i

Õ

ªo “ ¦ s ^z i = cos θ i   H    0 Au  i\  e ”   H “ ¦„  & h  Û ¼— 2 ; ~s i _ 

$ í

ì  r`  ¦
  · p . ô  Ǽ #  θ i   H F G y Œ •(polar angle) Õ ªo “ ¦ φ ⁱ



 H ~ ½ Ó0 Ay Œ •(azimuthal angle)`  ¦ y Œ •y Œ •
  · p . s  Û ¼— 2 ;>  _

 : Ÿ x > % i † < Æ& h  $ í | 9 “ É r ì  r C † < Êà º Z\  ¦ ½ ¨† < ÊÜ ¼– Ð" f % 3 # Q”  



. 7 £ ¤, a = βh x , b = βh y , c = βh z – Ð & ñ _  €  

Z(a, b, c, N ) = Z

e ^−βH dΩ =

N

Y

i=1

Z π 0

Z 2π 0

e ^{a sin θ}

ⁱ

^{cos φ}

ⁱ

^{+b sin θ}

ⁱ

^{sin φ}

ⁱ

^{+c cos θ}

ⁱ

sin θ i dθ i dφ i = [z i (a, b, c)] ^N . (15)

#

Œl " f

z i (a, b, c) = Z π

0

Z 2π 0

e ^{a sin θ}

ⁱ

^{cos φ}

ⁱ

^{+b sin θ}

ⁱ

^{sin φ}

ⁱ

^{+c cos θ}

ⁱ

sin θ i dθ i dφ i (16)

“

 X <, s  ~ ½ Ó& ñ d ” `  ¦ ~ ½ Ó& ñ d ”  (4)ü < q “ §K  ˜ Ѐ  

A = aˆ ~ x + bˆ y + cˆ z (17) s

Ù ¼– Ð

~

s i = s ^x _i x + s ˆ ^y _i y + s ˆ ^z _i z ˆ

= sin θ i cos φ i x + sin θ ˆ i sin φ i y + cos θ ˆ i z ˆ (18)



 H

ˆ

r = sin θ cos φˆ x + sin θ sin φˆ y + cos θˆ z (19)

\

 K { © œ† < Ê`  ¦ · ú ˜ à º e ”  .   " f ~ ½ Ó& ñ d ”  (13)Ü ¼– Ð Â Ò' 

z i (a, b, c) = 2π Z 1

−1

e

√

a

²

+b

²

+c

²

t dt,

= 4π

√ a ² + b ² + c ² sinh p

a ² + b ² + c ² . (20)

Õ ª QÙ ¼– Ð

Z(a, b, c, N ) = [z i (a, b, c)] ^N

=

 4π

√ a ² + b ² + c ² sinh( p

a ² + b ² + c ² )

 N

.

(21)

2. “ ¤” X ¢ä ì È ü w Š ¹ Å Heisenberg { ¢¨ | 

Á

ºô  Ç# 3 0 A “ ¦„   Heisenberg — ¸4 S q“ É r  6 £ § õ  ° ú  “ É r Hamil- tonian Ü ¼– Ð & ñ _ ô  Ç .

H = − J

4N (S _x ² + S _y ² + S _z ² ). (22) s

 d ” \ " f J  H Û ¼— 2 ;-Û ¼— 2 ;   ½ + Ë © œÃ ºs “ ¦ N“ É r 8 ú x Û ¼

—

2 ; > hà ºs  . 0 A_  Hamiltonian“ É r 8 ú x Û ¼— 2 ;$ í ì  r S α = P N

i=1 s ^α _i (α = x, y, z)\  ¦ s 6   x K " f ³ ð‰ & ³ % i   H X <, s ^α i   H   



& h  i\  0 Au ô  Ç é ß –{ 9  “ ¦„  Û ¼— 2 ;_  α-$ í ì  r s  . s  — ¸4 S q _

 ì  r C † < Êà º  H

Z H = Π ^N _i=1 Z π

0

Z 2π 0

sin θ i dθ i dφ i e

^4NβJ

^(S

^2x

^+S

^2y

^+S

^2z

⁾ (23) _

 + þ AI \  ¦ t   H X <,

e

^{γ2 +δ2 +ε24α}

= α π

Z ∞

−∞

e ^−αx

²

^+γx dx Z ∞

−∞

e ^−αy

²

^+δy dy Z ∞

−∞

e ^−αz

²

^+εz dz. (24)

\

 d ” \  α = N/βJ, γ = S ^x , δ = S y Õ ªo “ ¦ ε = S ^z \  ¦ @ /{ 9  €  

Z H (a, b, c) = ( N πβJ )

³²

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

e ⁻

^βJN

^(x

²

^+y

²

^+z

²

⁾ dxdydz

N

Y

i=1

Z π 0

Z 2π 0

sin θ i dθ i dφ i e ^{x sin θ}

ⁱ

^{cos φ}

ⁱ

^{+y sin θ}

ⁱ

^{sin φ}

ⁱ

^{+z cos θ}

ⁱ

= ( N πβJ )

³²

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

e ⁻

^βJN

^(x

²

^+y

²

^+z

²

⁾ [z _i (x, y, z)] ^N dxdydz (25)

e ”

Ü ¼– Ð ~ ½ Ó& ñ d ”  (13)s   r  s 6   x| ¨ c à º e ” 6 £ §`  ¦ · ú ˜ à º e ” 



.

3. w Š ¹ Å ­ Žä à ŠŒ ˜ m6 Kª Ž ®  o­ Ž { ¢¨ | 8 ý  ¹ ō ˜ mS ó o Þ

“

¦„   Û ¼— 2 ; ì ø ÍX <Ø Ô µ 1 ÏÛ ¼ — ¸4 S q“ É r [13–16] 8 ú x Û ¼— 2 ;$ í ì  r S α

(α = x, y, z) _  † < Êà º– Ð" f  6 £ § Hamiltonian

H = − J 4N

N

X

i6=j

[s ^x _i s ^x _j + s ^y _i s ^y _j ] − J z

4N

N

X

i6=j

s ^z _i s ^z _j (26)

Ü

¼– Ð l Õ ü t ÷ &  H X <, J ü < J z   H y Œ •y Œ • Û ¼— 2 ;-Û ¼— 2 ;   ½ + Ë © œÃ ºs 

“

¦, s ^α i (α = x, y, z)   H    & h  i\  Z  ~“   “ ¦„  Û ¼— 2 ;s  . F G y

Œ

•(polar angle) θ i ü < ~ ½ Ó0 Ay Œ •(azimuthal angle) φ i _  † ½ ÓÜ ¼

–

Ð" f ³ ð‰ & ³ €   y Œ •y Œ • s ^x i = sin θ i cos φ i , s ^y _i = sin θ i sin φ i , s ^z _i = cos θ i s  .

s

 — ¸4 S q_  „  í ß –r Ð 3 x`  ¦ à º' Ÿ    H · ú ˜“ ¦o 1 p u“ É r # Œ Qt 

 e ”   [17–19]. Û ¼— 2 ;s  3 " é ¶ Bloch ½ ¨³ ð€   © œ\ " f ƒ   5

Å q& h Ü ¼– Ð ¹ ¡ §f ” s l  M :ë  H \  \ P ×  æ „ ½ Ó · ú ˜“ ¦o 1 p u (heatbath algorithm) s  Metropolis · ú ˜“ ¦o 1 p u \  q K " f j þ t — ¸ e ” 



 [20]. s  · ú ˜“ ¦o 1 p u“ É r # Q‹ "  Û ¼— 2 ; C \ P `  ¦ | 9  S X ‰Ò  ¦ s  P (i = 1 · · · N |s ^x _i , s ^y _i , s ^z _i ) ∝ e ^−βH (27)

÷

&• ¸2 Ÿ ¤ Û ¼— 2 ; C \ P `  ¦ µ 1 ÏÒ q tr & ï  r  . 8 ú x Û ¼— 2 ;$ í ì  r S α \  ¦ S α = P N

i=1 s ^α _i (α = x, y, z) – Ð & ñ _   . €  $  j   P : Û ¼— 2 ;

“

 

~ s j = s ^x _j x + s ˆ ^y _j y + s ˆ ^z _j ˆ z (28) _

 D h\  v >  l  (update)\  ¦   H X <,
 Qt  Û ¼— 2 ;_  ½ + Ë`  ¦

S ~ ⁰ ≡

N

X

i6=j

(s ^x _i ˆ x + s ^y _i y + s ˆ ^z _i ˆ z

= S _x ⁰ x + S ˆ _y ⁰ y + S ˆ _z ⁰ z ˆ (29)



“ ¦ & ñ _   . Õ ª Q€   8 ú x Û ¼— 2 ;“ É r

S = ~ ~ S ⁰ + ~ s j (30)

–

Ð ³ ð‰ & ³ ) a  . s M : −βH   H Ó ü t o | ¾ ӓ É r d = βJ/4N s “ ¦ e = βJ z /4N  “ ¦ & ñ _  €  

−βH = d (S x ) ² + (S _y ) ²  + e(S z ) ² − d

N

X

i=1

(s ^x _i ) ² + (s ^y _i ) ²  − e

N

X

i=1

(s ^z _i ) ² = K + as ^x _j + bs ^y _j + cs ^z _j (31)

–

Ð ³ ð‰ & ³½ + É Ã º e ”   H X <, # Œl " f j   P : Û ¼— 2 ;_  D h\  v >  l \  ¦ à º' Ÿ r 

K = d[(S _x ⁰ ) ² + (S ⁰ _y ) ² ] + e(S ⁰ _z ) ² − d

N

X

i6=j

[(s ^x _i ) ² + (s ^y _i ) ² ] − e

N

X

i6=j

(s ^z _i ) ² (32)

ü

<

a = 2dS _x ⁰ , b = 2dS ⁰ _y , c = 2eS _z ⁰ (33)



 H    t  · ú §  H Ó ü t o | ¾ Ós  .   " f ~ ½ Ó& ñ d ”  (27)\  ~ ½ Ó& ñ d ”

 (31)`  ¦ @ /{ 9  €  

P (i = 1 · · · N |s ^x _i , s ^y _i , s ^z _i ) ∝ e ^K+as

^xj

^+bs

^yj

^+cs

^zj

(34)

  ) a  . Õ ª  X < \ P ×  æ „ ½ Ó · ú ˜“ ¦o 1 p u“ É r ² D G ™ è& h “   (local) · ú ˜

“

¦o 1 p u s Ù ¼– Ð, D h\  v >  l  à º' Ÿ r  K  H Ø  ¦ µ 1 Ï › ¸| ë ß –`  ¦ ï

 r  . Õ ªo  # Œ D h– Ðî  r j   P : Û ¼— 2 ;“ É r (s ^x _j ) ² + (s ^y _j ) ² + (s ^z _j ) ² = 1 _  ½ ¨5 Å q › ¸| `  ¦ ë ß –7 á ¤ # Œ  Ù ¼– Ð

P (s ^x _i , s ^y _i , s ^z _i ) = De ^as

^xj

^+bs

^yj

^+cs

^zj

δ[(s ^x _j ) ² + (s ^y _j ) ² + (s ^z _j ) ² − 1] (35)

_

 S X ‰Ò  ¦ – Ð" f µ 1 ÏÒ q t >   ) a  . # Œl " f D   H # Q‹ "  ½ ©

 

 o  © œÃ ºs  . 0 A_  S X ‰Ò  ¦ › ' a > d ” s  — ¸Ž  H Û ¼— 2 ;\  D

h\  v >  l  à º' Ÿ r  & h 6   x ÷ &Ù ¼– Ð j\  ¦ ~ ½ Ó& ñ d ” \ " f Ò q t| Ä Ì

“ ¦, Û ¼— 2 ; ° ú כ`  ¦ ½ ¨€  ý a³ ð>    à º θü < φ_  † ½ ÓÜ ¼– Ð  Ë ¨€  

P (θ, φ) = De a sin θ cos φ+b sin θ sin φ+c cos θ (36)

–

Ð ³ ð‰ & ³½ + É Ã º e ”  . # Œl " f ½ ©   o  © œÃ º D  H

Z π 0

Z 2π 0

P (θ, φ) sin θdθdφ = Z π

0

Z 2π 0

e a sin θ cos φ+b sin θ sin φ+c cos θ sin θdθdφ (37)

–

Ð Â Ò'  ½ ¨½ + É Ã º e ”   H X <, s  כ “ É r  r  ô  ǁ   ~ ½ Ó& ñ d ”  (13)õ 

° ú

 “ É r — ¸€ ª œ`  ¦ “ ¦ e ”  .

IV. + s Ç Â ] Ø

‘

: r ƒ  ½ ¨\ " f  H “ ¦„   Û ¼— 2 ;> _  ì  r C † < Êà º > í ß –\ " f


š

¸  H Z π 0

Z 2π 0

f (a sin θ cos φ + b sin θ sin φ + c cos θ) sin θdθdφ

_

 & h ì  r ° ú כ`  ¦ Û ¼º ú ˜ Y  L s  [ þ t # Q e ”   H † < Êà º_   â Ä º\   6   x

  H @ /g A$ í | 9  s 6   xZ O `  ¦  6   x K " f Ä »• ¸ % i  . Õ ªo “ ¦ s

 כ `  ¦ Ä »• ¸ l  0 A # Œ f ” ] X & h Ü ¼– Ð é ß –0 A 7 ˜' _   r„  

`

 ¦ à º' Ÿ  % i  . Õ ªo “ ¦ s  & h ì  r s  # Œ Q “ ¦„   Û ¼— 2 ;> _  ƒ  

½

¨\  ´ ú §s  6 £ x6   xH † d`  ¦ ˜ Ð% i  .

P

c p 8 ý ò k >

s

  7 Hë  H“ É r 2011¸  • ¸ Ø  æ· ¡ ¤ @ /† < Ɠ § † < ÆÕ ü tƒ  ½ ¨t " é ¶  \ O _ 

ƒ

 ½ ¨q  t " é ¶ \  _  # Œ ƒ  ½ ¨÷ &% 3 _ þ v m  .   ¨ 8 Š ~ ½ Ó& ñ d ”  (1) _

 Ä »• ¸\  › ' a ô  Ç s " é ¶ d ”  “ §Ã º_ ” _  “ ¦| \  y Œ ™ × ¼w n m  .

Y

c p w Š à U Ø ”  ô

[1] C. Domb and M. S. Green, Phase Transitions and Critical Phenomena (Academic Press, London, 1972), Vol. 1.

[2] G. M. Bell and D. A. Lavis, Statistical Mechanics of Lattice Models (Ellis Horwood. Chichester, 1989), Vol. 1.

[3] B. M. McCoy and T. T. Wu, The Two-Dimensional

Ising Model (Harvard Univ. Press, Cambridge

Mass., 1973).

[4] R. J. Baxter, Exactly Solved Models in Statisitcal Mechanics (Academic Press, New York, 1989).

[5] M. Takahashi, Thermodynamics of One- Dimensional Solvable Models (Cambridge Univ.

Press, Cambridge, 1999).

[6] A. P. Prudnikov, Yu. A. Brychkov and O. I.

Marichev, Integrals and Series (Gordon and Breach, New York, 1986), Vol. 1, p. 573.

[7] I. S. Gradshteyn and I. M. Ryzhik, Table of Inte- grals, Series and Products (Academic Press, New York, 1965), p. 620.

[8] J. Mathews and R. L. Walker, Mathematical Meth- ods of Physics ( Benjamin, Menlo Park, 1970), p.

62.

[9] R. W. Brockett in Mathematical Theory of Networks and Systems (Proceedings of the International Sym- posium held at the Ben Gurion University of the Negev, Beer Sheva, June 20-24, 1983) edited by P.

A. Fuhrmann (Springer-Verlag, Berlin, 1984) [10] R. M. Murray, Z. Li and S. S. Sastry, A Mathe-

matical Introduction to Robotic Manipulation (CRC Press, Boca Raton, 1994).

[11] E. W. Weisstein, “Rodrigues’ Rotation Formula”

from MathWorld, http://mathworld.wolfram.com/.

[12] If we are not interested in obtaining a relationship between the original coordinated system and the

new coordinated system, the derivation can be sim- plified. In Eq. (4), since ~ A · ˆ r is a scalar, it is in- variant under any coordinate transformation. Hence, if ~ A ⁰ = Aˆ z in the new coordinate system, then A ~ ⁰ · ˆ r ⁰ = A cos θ ⁰ and dΩ ⁰ = sin θ ⁰ dθ ⁰ dφ ⁰ . Thereby, if we set cos θ ⁰ ≡ t, then dΩ ⁰ = dtdφ ⁰ and Eq. (13) can be obtained right away.

[13] R. Dekeyser and M. H. Lee, Phys. Rev. B 19, 265 (1979).

[14] S. K. Oh, New Physics: Sae Mulli 60, 622 (2010).

[15] S. K. Oh, J. Korean Phys. Soc. 23, 485 (1990).

[16] S. K. Oh, C. N. Yoon and J. S. Chung, Phys. Rev.

B 53, 11 537 (1996).

[17] D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge Univ. Press, Cambridge, UK, 2000 ).

[18] M. E. J. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics (Clarendon Press, Oxford, UK, 1999).

[19] W. Krauth, Statistical Mechanics: Algorithms and Computations (Oxford Univ. Press, Oxford, UK, 2006).

[20] S. K. Oh and H. W. Lee, in Progress of Statistical

Physics, edited by I. M. Kim and D. Kim (Minumsa,

Seoul, 1990), Vol V.